For playing First Person Shooter games, the demand for the pointer control is exactly that, fast jump to n points, meaning, straight lines movements of the mouse.

Who Prefer Trackball?

Many 3D Modeling software users prefer trackball over mouse. For more intuitive control and rotation of objects or change viewing angle.

Some gamers prefer trackball.

Disabled or persons with Repetitive Strain Injury, prefer trackball.

How to Choose a Trackball?

Most Important: Comfort

The most important thing when buying a trackball is comfort. How does it physically fit your hand.

All other aspects of trackball are less important than comfort.

Trackball's comfort is most important, because, it is this aspect you have to live with. When a trackball's ball size, position, button position, are designed in a way that you find it hard to operate, then it is a major pain.

Thumb vs Index Finger Operation

Trackballs have many different designs and shapes. One major
difference is thumb-operated ball vs middle-finger or pointing-finger
operated ball.

Trackballs designed for index finger is easier to operate than the
thumb. Because, with index finger design, you can use the whole hand
to move the ball. Thumb based design usually have the ball on the
side, and you must use thumb.

Does the Ball Spin?

When you buy a trackball, the less the ball friction, the better. If you can make the ball spin for second or more, it's GOOD.

But, ultimately, it doesn't matter much. Because you can set acceleration in your Operating System {Microsoft Windows, Linux, Mac}. That is, with acceleration, the pointer moves further when you roll the ball faster.